MannWhitneyUTest.java
- /*
- * Licensed to the Apache Software Foundation (ASF) under one or more
- * contributor license agreements. See the NOTICE file distributed with
- * this work for additional information regarding copyright ownership.
- * The ASF licenses this file to You under the Apache License, Version 2.0
- * (the "License"); you may not use this file except in compliance with
- * the License. You may obtain a copy of the License at
- *
- * http://www.apache.org/licenses/LICENSE-2.0
- *
- * Unless required by applicable law or agreed to in writing, software
- * distributed under the License is distributed on an "AS IS" BASIS,
- * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
- * See the License for the specific language governing permissions and
- * limitations under the License.
- */
- package org.apache.commons.math4.legacy.stat.inference;
- import org.apache.commons.statistics.distribution.NormalDistribution;
- import org.apache.commons.math4.legacy.exception.ConvergenceException;
- import org.apache.commons.math4.legacy.exception.MaxCountExceededException;
- import org.apache.commons.math4.legacy.exception.NoDataException;
- import org.apache.commons.math4.legacy.exception.NullArgumentException;
- import org.apache.commons.math4.legacy.stat.ranking.NaNStrategy;
- import org.apache.commons.math4.legacy.stat.ranking.NaturalRanking;
- import org.apache.commons.math4.legacy.stat.ranking.TiesStrategy;
- import org.apache.commons.math4.core.jdkmath.JdkMath;
- import java.util.stream.IntStream;
- /**
- * An implementation of the Mann-Whitney U test (also called Wilcoxon rank-sum test).
- *
- */
- public class MannWhitneyUTest {
- /** Ranking algorithm. */
- private NaturalRanking naturalRanking;
- /**
- * Create a test instance using where NaN's are left in place and ties get
- * the average of applicable ranks. Use this unless you are very sure of
- * what you are doing.
- */
- public MannWhitneyUTest() {
- naturalRanking = new NaturalRanking(NaNStrategy.FIXED,
- TiesStrategy.AVERAGE);
- }
- /**
- * Create a test instance using the given strategies for NaN's and ties.
- * Only use this if you are sure of what you are doing.
- *
- * @param nanStrategy
- * specifies the strategy that should be used for Double.NaN's
- * @param tiesStrategy
- * specifies the strategy that should be used for ties
- */
- public MannWhitneyUTest(final NaNStrategy nanStrategy,
- final TiesStrategy tiesStrategy) {
- naturalRanking = new NaturalRanking(nanStrategy, tiesStrategy);
- }
- /**
- * Ensures that the provided arrays fulfills the assumptions.
- *
- * @param x first sample
- * @param y second sample
- * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
- * @throws NoDataException if {@code x} or {@code y} are zero-length.
- */
- private void ensureDataConformance(final double[] x, final double[] y)
- throws NullArgumentException, NoDataException {
- if (x == null ||
- y == null) {
- throw new NullArgumentException();
- }
- if (x.length == 0 ||
- y.length == 0) {
- throw new NoDataException();
- }
- }
- /** Concatenate the samples into one array.
- * @param x first sample
- * @param y second sample
- * @return concatenated array
- */
- private double[] concatenateSamples(final double[] x, final double[] y) {
- final double[] z = new double[x.length + y.length];
- System.arraycopy(x, 0, z, 0, x.length);
- System.arraycopy(y, 0, z, x.length, y.length);
- return z;
- }
- /**
- * Computes the <a
- * href="http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U"> Mann-Whitney
- * U statistic</a> comparing mean for two independent samples possibly of
- * different length.
- * <p>
- * This statistic can be used to perform a Mann-Whitney U test evaluating
- * the null hypothesis that the two independent samples has equal mean.
- * </p>
- * <p>
- * Let X<sub>i</sub> denote the i'th individual of the first sample and
- * Y<sub>j</sub> the j'th individual in the second sample. Note that the
- * samples would often have different length.
- * </p>
- * <p>
- * <strong>Preconditions</strong>:
- * <ul>
- * <li>All observations in the two samples are independent.</li>
- * <li>The observations are at least ordinal (continuous are also ordinal).</li>
- * </ul>
- *
- * @param x the first sample
- * @param y the second sample
- * @return Mann-Whitney U statistic (minimum of U<sup>x</sup> and U<sup>y</sup>)
- * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
- * @throws NoDataException if {@code x} or {@code y} are zero-length.
- */
- public double mannWhitneyU(final double[] x, final double[] y)
- throws NullArgumentException, NoDataException {
- ensureDataConformance(x, y);
- final double[] z = concatenateSamples(x, y);
- final double[] ranks = naturalRanking.rank(z);
- double sumRankX = 0;
- /*
- * The ranks for x is in the first x.length entries in ranks because x
- * is in the first x.length entries in z
- */
- sumRankX = IntStream.range(0, x.length).mapToDouble(i -> ranks[i]).sum();
- /*
- * U1 = R1 - (n1 * (n1 + 1)) / 2 where R1 is sum of ranks for sample 1,
- * e.g. x, n1 is the number of observations in sample 1.
- */
- final double u1 = sumRankX - ((long) x.length * (x.length + 1)) / 2;
- /*
- * It can be shown that U1 + U2 = n1 * n2
- */
- final double u2 = (long) x.length * y.length - u1;
- return JdkMath.min(u1, u2);
- }
- /**
- * @param umin smallest Mann-Whitney U value
- * @param n1 number of subjects in first sample
- * @param n2 number of subjects in second sample
- * @return two-sided asymptotic p-value
- * @throws ConvergenceException if the p-value can not be computed
- * due to a convergence error
- * @throws MaxCountExceededException if the maximum number of
- * iterations is exceeded
- */
- private double calculateAsymptoticPValue(final double umin,
- final int n1,
- final int n2)
- throws ConvergenceException, MaxCountExceededException {
- /* long multiplication to avoid overflow (double not used due to efficiency
- * and to avoid precision loss)
- */
- final long n1n2prod = (long) n1 * n2;
- // http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U#Normal_approximation
- final double eU = n1n2prod / 2.0;
- final double varU = n1n2prod * (n1 + n2 + 1) / 12.0;
- final double z = (umin - eU) / JdkMath.sqrt(varU);
- // No try-catch or advertised exception because args are valid
- // pass a null rng to avoid unneeded overhead as we will not sample from this distribution
- final NormalDistribution standardNormal = NormalDistribution.of(0, 1);
- return 2 * standardNormal.cumulativeProbability(z);
- }
- /**
- * Returns the asymptotic <i>observed significance level</i>, or <a href=
- * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">
- * p-value</a>, associated with a <a
- * href="http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U"> Mann-Whitney
- * U statistic</a> comparing mean for two independent samples.
- * <p>
- * Let X<sub>i</sub> denote the i'th individual of the first sample and
- * Y<sub>j</sub> the j'th individual in the second sample. Note that the
- * samples would often have different length.
- * </p>
- * <p>
- * <strong>Preconditions</strong>:
- * <ul>
- * <li>All observations in the two samples are independent.</li>
- * <li>The observations are at least ordinal (continuous are also ordinal).</li>
- * </ul><p>
- * Ties give rise to biased variance at the moment. See e.g. <a
- * href="http://mlsc.lboro.ac.uk/resources/statistics/Mannwhitney.pdf"
- * >http://mlsc.lboro.ac.uk/resources/statistics/Mannwhitney.pdf</a>.</p>
- *
- * @param x the first sample
- * @param y the second sample
- * @return asymptotic p-value
- * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
- * @throws NoDataException if {@code x} or {@code y} are zero-length.
- * @throws ConvergenceException if the p-value can not be computed due to a
- * convergence error
- * @throws MaxCountExceededException if the maximum number of iterations
- * is exceeded
- */
- public double mannWhitneyUTest(final double[] x, final double[] y)
- throws NullArgumentException, NoDataException,
- ConvergenceException, MaxCountExceededException {
- ensureDataConformance(x, y);
- final double uMin = mannWhitneyU(x, y);
- return calculateAsymptoticPValue(uMin, x.length, y.length);
- }
- }